I'm new to magic the gathering in general, and I've just started playing it on steam. I keep seeing references to "kicked" cards. How does that work?
It likely refers to the keyword ability "Kicker". It allows paying an additional cost as one casts a spell in order to get some effect when the spell resolves.
For example, Rite of Replication is a Sorcery with a cost of 2 blue mana and two colorless mana, as well as a Kicker cost of 5 colorless mana and has the text, "Put a token onto the battlefield that's a copy of target creature. If Rite of Replication was kicked, put 5 of those tokens onto the battlefield instead."
Thus, in that particular example, if one can play the spell for 2 blue and 7 colorless, they'll get a significantly larger effect (5 tokens instead of 1).